Stutter can be caused by a busy server or if the video has a bitrate which is too high for the Chromecast to keep up. Especially on weekends the content providers are overloaded and can't provide the videos fast enough. If possible try a SD stream instead of HD.
The Chromecast can't buffer some streams at all and has a very small buffer. this might get fixed with a future Chromecast firmware update.

Bottleneck - Local network streams

Don't try to stream Blue-ray quality movies over WiFi. E.g. the Chromecast only supports 2.4GHz which is not enough to stream those movies. It won't work.

Bitrate: Some phones shoot videos with a really high bitrate. See above.

The Chromecast 1 or the Amazon Fire TV Stick both come with a HDMI extender cord and with a extra power supply unit.

USE BOTH!

The extra juice from the power supply (instead of plugging the usb cord into your TV) can improve the WiFi signal. It brings the dongle further away from the TV, which can improve the signal.

1. Make sure that all your devices are on the same network.
2. Turn WiFi off and on again on your streaming device. Wait a couple of second and check if it appears in the list.
3. Swipe LocalCast away in the multi tasking menu - reopen it. Wait a couple of second and check if it appears in the list.
4. Reboot all your things (Router, streaming device, receiving device)

Xbox 360:
Go to Settings: Turn "Play To" on
If it doesn't work: Turn "Play To" off and on

Xbox One:
Install Xbox Video and Xbox Music
Got Xbox One Settings: Turn "Play To" on
If it doesn't work: Turn "Play To" off and on
Playing a game isn't possible while casting.

This means that your device (Chromecast, Smart TV, ...) doesn't support the audio codec of your video.
All receiving devices support a couple of codecs and most video formats.
For example the Chromecast's supported formats and codecs are listed here.

Video files are containers which hold a video and a audio file. If only one of both plays, then the other one uses a codec which is not supported by your receiving device. To convert the files on your PC I recommend www.handbrake.fr

This means that your device (Chromecast, Smart TV, ...) doesn't support the video codec of your video.
All receiving devices support a couple of codecs and most video formats.
For example the Chromecast's supported formats and codecs are listed here.

Video files are containers which hold a video and a audio file. If only one of both plays, then the other one uses a codec which is not supported by your receiving device. To convert the files on your PC I recommend www.handbrake.fr

This happens when the play store app doesn't report that there are purchases and this can have multiple reasons(, most likely because you have multiple Google accounts on your phone or if your Google Wallet account isn't set up properly.):

Uninstall LocalCast
In a BROWSER on your desktop: Log in to the web interface of Google Play with the account you used to purchase
Install LocalCast from the web interface to your device. If you have no desktop PC: Don't let it open the Playstore app, but use the install button and choose your device in the browser.

There are multiple price options, because I want you to decide how much the app is worth to you.
All purchases unlock the Pro Version, there is no difference, besides the price.
There are multiple one time purchases and subscriptions ("/month" or "/year").
If you aren't sure what the purchase is, just touch it. The official Google Playstore dialog opens, which should make things clearer.

DLNA is a direct connection to your cast device.
SMB has to be routed through the phone/tablet, which doubles the needed bandwidth.
Route audio doesn't work with SMB.
If possible choose a DLNA server like Serviio.